This paper considers the problem of searching over a collection of sequences that are generated by one of the two possible distributions $F_0$ and $F_1$ and designs the quickest sequential detection procedure for identifying one sequence that is generated according to $F_1$. Generation of the sequences obeys a known dependency kernel such that the prior probability that a sequence is generated by $F_1$ is governed by the distributions of the rest of sequences. The optimal quickest sequential detection procedure, that is the procedure that strikes a balance between detection quality and decision delay as two opposing performance measures, is shown to have a sequential probability ratio test (SPRT) structure.